6.1 midpoints and perpendicular bisectors, 6.2 equation of a circle

How can you find the midpoint of a line segment?

By averaging the x and y coordinates of its endpoints

Example 1

Example 2

What is the perpendicular bisector of a line segment?

The straight line that is perpendicular to the line and passes through the midpoint of the line

If the gradient of AB is m, then the gradient of the it’s perpendicular bisector will be

-1/m

Example 3

What is the equation of a circle with centre (0,0)

x² + y² = r²

What is the equation of a circle with centre (a,b)?

(x-a)² + (y-b)² = r²

Example 4

Example 5

Example 6

You can multiply out the brackets in the equation of a circle to find it in an alternative form

What form can the equation of a circle be given in after expanding?

x² + y² + 2fx + 2gy + c = 0

What centre and radius would this have:

x² + y² + 2fx + 2gy + c = 0

Centre: (-f, -g)

Radius: square root of (f² + g²- c)

To find the centre and radius of a circle with an equation given in expanded form, it is best to

Complete the square for the x and y terms

Example 7